Research on reinforcement learning based on PPO algorithm for human-machine intervention in autonomous driving

1.   Introduction

This research focuses on the comprehensive connection among analytic functions and their inverses, which provides new ideas for investigating coefficient estimates and inequalities. The outcome of the present study is particularly relevant in the framework of geometric function theory (GFT), where particular geometric properties are established for analytic functions employing methods specific to this domain of research, but also could offer applications in other related fields such as partial differential equation theory, engineering, fluid dynamics, and electronics. Tremendous impact in the development of GFT was given by the Bieberbach's conjecture, an essential problem related to coefficient estimates for functions that lie within the family $\mathcal{S}$ of univalent functions. This conjecture suggests that for $f\in \mathcal{S}$, expressed through the Taylor–Maclaurin series expansion:

where $\mathbb{D}: = \left \{ \upsilon \in \mathbb{C}:\left \vert \upsilon \right \vert < 1\right \},$ the coefficients inequality $\left \vert d_{k}\right \vert \leq k$ holds for all $k\geq 2.$ The family of such analytic functions with the series representation provided in $\left(1.1\right)$ is represented by $\mathcal{A}$. It is worth mentioning that Koebe first introduced $\mathcal{S}$ as a subclass of $\mathcal{A}$ in 1907. Bieberbach ^[1] originally proposed this conjecture in 1916, initially verifying it for the case $k = 2$. Subsequent advancements by researchers including Löwner ^[2], Garabedian and Schiffer ^[3], Pederson and Schiffer ^[4], and Pederson ^[5] offered partial proofs for cases up to $k = 6$. However, the complete proof for $k\geq 7$ remained unsolved until 1985, when de-Branges ^[6] utilized hypergeometric functions to establish it for $k\geq 2$.

In 1960, Lawrence Zalcman postulated the functional $\left \vert d_{k}^{2}-d_{2k-1}\right \vert \leq \left(k-1\right) ^{2}$ with $k\geq 2$ for $f\in \mathcal{S}$ in order to establish the Bieberbach hypothesis. This has led to the publication of several papers ^[7,8,9] on the Zalcman hypothesis and its generalized form $\left \vert \lambda d_{k}^{2}-d_{2k-1}\right \vert \leq \lambda k^{2}-2k+1$ with $\lambda \geq 0$ for various subfamilies of the family $\mathcal{S}$. This hypothesis remained unproven for a long time until Krushkal's breakthrough in 1999, when he proved it in ^[10] for $k\leq 6$ and solved it by utilizing the holomorphic homotopy of univalent functions in an unpublished manuscript ^[11] for $k\geq 2.$ It was also demonstrated that $\left \vert d_{k}^{l}-d_{2}^{l\left(k-1\right) }\right \vert \leq 2^{l\left(k-1\right) }-k^{l}$ with $k, l\geq 2$ for $f\in \mathcal{S}$. The Bieberbach conjecture landscape is further enhanced by other conjectures, such as the one presented by Ma ^[12] in 1999, which is

He restricted his proof to a subclass of $\mathcal{S}$. The challenge for class $\mathcal{S}$ remains available.

Now, let us recall the concept of subordination, which essentially describes a relationship between analytic functions. An analytic function $g_{1}$ is subordinate to $g_{2}$ if there exists a Schwarz function $\omega$ such that $g_{1}(\upsilon) = g_{2}\left(\omega (\upsilon)\right)$ and it is mathematically represented as $g_{1}\prec g_{2}.$ If $g_{2}$ is univalent in $\mathbb{D}$, then

if and only if

In essence, this relationship helps us understand how one function is "contained" within another, providing insights into their behavior within the complex plane. The family of univalent functions comprises three classic subclasses $\mathcal{C},$ $\mathcal{S}^{\ast },$ and $\mathcal{K}$, each distinguished by its unique properties. These subclasses are commonly known as convex functions, starlike functions, and close-to-convex functions, respectively. Let us define each class:

and

for some $h\in \mathcal{S}^{\ast }$. The above family $\mathcal{K}$ may be reduced to the family of bounded turning functions $\mathcal{BT}$ by choosing $h\left(\upsilon \right) = \upsilon.$ Moreover, a number of intriguing subfamilies of class $\mathcal{S}$ were examined by replacing $\frac{1+\upsilon }{1-\upsilon }$ by other special functions. For the reader's benefit, a few of them are included below:

$\left(i\right)$. $\mathcal{S}_{e}^{\ast }\equiv \mathcal{S}^{\ast }\left(e^{z}\right)$ and $\mathcal{C}_{e}\equiv \mathcal{C}\left(e^{z}\right)$ ^[13]$,$ $\mathcal{S}_{SG}^{\ast }\equiv \mathcal{S} ^{\ast }\left(\frac{2}{1+e^{-z}}\right)$ and $\mathcal{C}_{SG}\equiv \mathcal{C}\left(\frac{2}{1+e^{-z}}\right)$^[14],

$\left(ii\right)$. $\mathcal{S}_{cr}^{\ast }\equiv \mathcal{S} ^{\ast }\left(z+\sqrt{1+z^{2}}\right)$ and $\mathcal{C}_{cr}\equiv \mathcal{C}\left(z+\sqrt{1+z^{2}}\right)$ ^[15]$,$ $\mathcal{S} _{Ne}^{\ast }\equiv \mathcal{S}^{\ast }\left(1+z-\frac{1}{3}z^{3}\right)$ ^[16],

$\left(iii\right)$. $\mathcal{S}_{(n-1)\mathcal{L}}^{\ast }\equiv \mathcal{S}^{\ast }(\Psi _{n-1}\left(z\right))$ ^[17] with $\Psi _{n-1}\left(z\right) = 1+\frac{n}{n+1}z+\frac{1}{n+1}z^{n}$ for $n\geq 2$.

$\left(iv\right)$. $\mathcal{S}_{\sinh }^{\ast }\equiv \mathcal{S} ^{\ast }\left(1+\sinh (\lambda z)\right)$ with $0\leq \lambda \leq \ln \left(1+\sqrt{2}\right)$ ^[18].

It is observed that a significant area of mathematics is the study of the inverse functions for the functions in various subclasses of $\mathcal{S}$. The well-known Koebe's $1/4$ theorem states that there exists the inverse $f^{-1}$ for every univalent function $f$ defined in $\mathbb{D}$, at least on the disk with a radius of 1/4, which has Taylor's series form

Employing the formula $f\left(f^{-1}\left(\omega \right) \right) = \omega,$ we acquire

We consider the Hankel determinant of $f^{-1}$ given by

Specifically, the second and third-order Hankel determinants of $f^{-1}$ are defined as the following determinants, respectively:

As it is seen, $f^{-1}$ is also not necessary to be univalent. Thus, this concept is also a natural generalization of the Hankel determinant with coefficients of $f\in \mathcal{S}$ as entries. There are very few publications in the literature that address coefficient-related problems of the inverse function, particularly determinants as stated above. Due to such a reason, the researchers motivated, and so this led to the publication of some good articles ^{[19,20,21,22,23]} on the above-stated Hankel determinants.

The key mathematical concept in this study is the Hankel determinant $\hat{H} _{\lambda, n}\left(f\right)$, where $n, \lambda \in \{1, 2, \ldots \}.$ This concept was introduced by Pommerenke ^[24,25]. It is composed of the coefficients of the function $f\in \mathcal{S}$ and is expressed as

This determinant is utilized in both pure mathematics and applied sciences, including non-stationary signal theory in the Hamburger moment problem, Markov process theory, and a variety of other fields. There are relatively few publications on the estimates of the Hankel determinant for functions in the general class $\mathcal{S}$. Hayman established the best estimate for $f\in \mathcal{S}$ in ^[26] by asserting that $\left \vert \hat{H} _{2, n}\left(f\right) \right \vert \leq \left \vert \eta \right \vert$, where $\eta$ is a constant. Moreover, it was demonstrated in ^[27] that $\left \vert \hat{H}_{2, 2}\left(f\right) \right \vert \leq \eta$ where $0\leq \eta \leq 11/3$ for $f\in \mathcal{S}$. The two determinants $\hat{H} _{2, 1}\left(f\right)$ and $\hat{H}_{2, 2}\left(f\right)$ for different subfamilies of univalent functions have been thoroughly examined in the literature. Notable work was done by Janteng et al. ^[28], Lee et al. ^[29], Ebadian et al. ^[30], and Cho et al. ^[31], who determine the sharp estimates of the second-order Hankel determinant for certain subclasses of $\mathcal{S}$.

The sharp estimate of the third-order Hankel determinant $\hat{H} _{3, 1}\left(f\right)$ for some analytic univalent functions is mathematically more difficult to find than the second-order Hankel determinant. Numerous articles on the third-order Hankel determinant have been published in the literature in which nonsharp limits of this determinant for the fundamental subclasses of analytic functions are determined. Following these arduous investigations, a few scholars were eventually able to obtain sharp bounds of this determinant for the classes $\mathcal{C},$ $\mathcal{BT},$ and $\mathcal{S}^{\ast }$, as reported in the recently published works ^[32,33,34], respectively. These estimates are given by

Later on, Lecko et al. ^[35] established the sharp estimate for $\left \vert \hat{H}_{3, 1}\left(f\right) \right \vert$ by utilizing similar approaches, specifically for functions that belong to the $\mathcal{S }^{\ast }\left(1/2\right)$ class. Also, the articles ^[36,37,38] provide more investigations on the exact bounds of this third-order Hankel determinant.

Now, let us consider the three function classes defined respectively by

and

These classes have been studied by Faisal et al. ^[39], Tang et al. ^[40], and Mendiratta et al. ^[13] respectively. In this paper, we improved the bound of the third-order Hankel determinant $\left \vert \hat{H}_{3, 1}\left(f^{-1}\right) \right \vert$, which was determined by Hu and Deng ^[41] and published recently in AIMS Mathematics. Furthermore, we obtain the bounds of the initial three inverse coefficients together with the sharp bounds of Krushkal, Zalcman, and Fekete–Szeg ö functionals along with the Hankel determinants $\left \vert \hat{H} _{2, 2}\left(f^{-1}\right) \right \vert$ and $\left \vert \hat{H} _{3, 1}\left(f^{-1}\right) \right \vert$ upper bounds.

2.   A set of lemmas

Let $\mathcal{B}_{0}$ be the class of Schwarz functions. It is noted that $\omega \in \mathcal{B}_{0}$ can be written as

We require the following lemmas to prove our main results.

Lemma 2.1. ^[42] Let $\omega \left(\upsilon \right)$ be a Schwarz function. Then, for any real numbers $\varrho$ and $\varsigma$ such that

the following sharp estimate holds:

Lemma 2.2. ^[43] If $\omega \left(\upsilon \right)$ be a Schwarz function$,$ then

Moreover, for $\tau \in \mathbb{C}$, the following inequality holds

Lemma 2.3. ^[44] Let $\omega \left(\upsilon \right)$ be a Schwarz function. Then

Lemma 2.4. ^[45] Let $\omega \left(\upsilon \right)$ be a Schwarz function. Then

and

3.   Third-order Hantel determinant

In this section, we will improve the bound of the third-order Hankel determinant $\left \vert \hat{H}_{3, 1}\left(f^{-1}\right) \right \vert$ with inverse coefficient entries for functions belonging to the class $\mathcal{S}_{\mathfrak{Sg}}^{\ast }.$

Theorem 3.1. Let $f^{-1}$ be the inverse of the function $f\in \mathcal{S}_{ \mathfrak{Sg}}^{\ast }$ and has the form $\left(1.2\right)$. Then

Proof. Let $f\in \mathcal{S}_{\mathfrak{Sg}}^{\ast }.$ Then, by subordination relationship, it implies

and also assumes that

Using $\left(1.1\right)$, we obtain

By some easy calculation and utilizing the series expansion of $\left(3.2\right)$, we achieve

Now by comparing $\left(3.3\right)$ and $\left(3.4\right)$, we obtain

Putting $\left(3.5\right)$–$\left(3.8\right)$ in $\left(1.3\right)$–$\left(1.6\right)$, we obtain

The determinant $\left \vert \hat{H}_{3, 1}\left(f^{-1}\right) \right \vert$ can be reconfigured as follows:

From $\left(3.9\right)$–$\left(3.12\right)$, we easily write

Now we begin by utilizing Lemma 2.1 with $\varrho = -\frac{1}{2}$ and $\varsigma = -\frac{1}{6}$ that

and also by using Lemma 2.3, we have

Applying it and also using $\left \vert \sigma _{4}\right \vert \leq 1-\left \vert \sigma _{2}\right \vert ^{2}-\left \vert \sigma _{1}\right \vert ^{2},$ we achieve

where

But $E$ is a decreasing function of the variable $\sigma;$ consequently,

The function $E\left(0, t\right)$ reaches its maximum value in $\left[0, 1 \right]$ if $t = -\frac{1}{3}+\frac{1}{3}\sqrt{13},$ so $E\left(0, t\right) \leq 2.0322,$ which completes the proof. □

Conjecture 3.2. If the inverse of $f\in \mathcal{S}_{\mathfrak{Sg}}^{\ast }$ is of the form $\left(1.2\right)$, then

Equality will be obtained by using $\left(1.3\right)$–$\left(1.6\right)$ together with

4.   Inverse coefficients for $\mathcal{S}_{3l, s}^{\ast }$

We begin this section by computing the estimates of the first three initial inverse coefficients for functions in the family $\mathcal{S}_{3l, s}^{\ast }.$

Theorem 4.1. Let the inverse function of $f\in \mathcal{S}_{3l, s}^{\ast }$ has the series form $\left(1.2\right)$. Then

The equality can easily be obtained by utilizing $\left(1.3\right)$ up to $\left(1.5\right)$ together with

Proof. Let $f\in \mathcal{S}_{3l, s}^{\ast }.$ Then we easily write

and here $\omega$ represents the Schwarz function. Also, let us assume that

Using $\left(1.1\right),$ we obtain

By some easy calculation and utilizing the series expansion of $\left(4.2\right)$, we have

Now, by comparing $\left(4.3\right)$ and $\left(4.4\right)$, we obtain

Substituting $\left(4.5\right)$–$\left(4.8\right)$ in $\left(1.3\right)$–$\left(1.6\right)$, we obtain

Using $\left(2.2\right)$ in $\left(4.9\right)$, we achieve

To prove the second inequality, we can write $\left(4.10\right)$ as

Applying $\left(2.3\right)$ in the above equation, we achieve

From $\left(4.11\right),$ we deduce that

Comparing it with Lemma 2.1, we note that

It is clear that $2\leq \left \vert \varrho \right \vert \leq 4$ with

All the conditions of Lemma 2.1 are satisfied. Therefore

The required proof is thus completed. □

Now, we compute the Fekete–Szegö functional bound for the inverse function of $f\in \mathcal{S}_{3l, s}^{\ast }.$

Theorem 4.2. If $f^{-1}$ is the inverse of the function $f\in \mathcal{S} _{3l, s}^{\ast }$ with series expansion $\left(1.2\right)$, then

This functional bound is sharp.

Proof. Putting $\left(4.9\right)$ and $\left(4.10\right)$, we obtain

Application of $\left(2.3\right)$ leads us to

The bound of the above functional is best possible, and it can easily be checked by $\left(1.3\right)$, $\left(1.4\right)$, and $\left(4.1\right)$ with $m = 2.$ □

By replacing $\tau = 1$ in Theorem 4.2, we arrive at the below result.

Corollary 4.3. If the inverse of the function $f\in \mathcal{S}_{3l, s}^{\ast }$ is $f^{-1}$ with series expansion $\left(1.2\right)$, then

This estimate is sharp, and equality will be obtained by using $\left(1.3\right)$, $\left(1.4\right)$, and $\left(4.1\right)$ with $m = 2.\ $

Next, we investigate the Zalcman functional upper bound for $f^{-1}\in \mathcal{S} _{3l, s}^{\ast }.$

Theorem 4.4. If $f\in \mathcal{S}_{3l, s}^{\ast }$ and its inverse function $f^{-1}$ have the form $\left(1.2\right)$, then

The above estimate is sharp.

Proof. Taking use of $\left(4.9\right)$–$\left(4.11\right),$ we achieve

From Lemma 2.1, let

It is clear that $2\leq \left \vert \varrho \right \vert \leq 4$ with

Thus, all the conditions of Lemma 2.1 are satisfied. Hence

The required estimate is best possible and will easily be obtained by using $\left(1.3\right)$–$\left(1.5\right)$, and $\left(4.1\right)$ with $m = 3.$ □

Further, we intend to compute the Krushkal functional bound for the family $\mathcal{S}_{3l, s}^{\ast }.$

Theorem 4.5. If $f\in \mathcal{S}_{3l, s}^{\ast }$ and its inverse function $f^{-1}$ have the form $\left(1.2\right)$, then

This estimate is sharp.

Proof. Putting $\left(4.9\right)$ and $\left(4.11\right)$, we obtain

From Lemma 2.1, let

It is clear that $2\leq \left \vert \varrho \right \vert \leq 4$ with

Thus, all the conditions of Lemma 2.1 are satisfied. Hence

This estimate is best possible and will be confirmed by using $\left(1.3\right)$, $\left(1.5\right)$, and $\left(4.1 \right)$ with $m = 3.$ □

In the upcoming result, we will investigate the estimate of $\hat{H}_{2, 2}\left(f^{-1}\right)$ for the family $\mathcal{S}_{3l, s}^{\ast }.$

Theorem 4.6. Let the inverse function of $f\in \mathcal{S}_{3l, s}^{\ast }$ has the series expansion $\left(1.2\right)$. Then

This inequality is sharp, and equality will easily be achieved by using $\left(1.3\right)$–$\left(1.5\right)$ and $\left(4.1\right)$ with $m = 2.$

Proof. The determinant $\hat{H}_{2, 2}\left(f^{-1}\right)$ can be reconfigured as follows:

Substituting $\left(4.9\right)$–$\left(4.11\right)$, we achieve

where

and

Utilizing Lemma 2.4, we acquire $Y_{1}\leq 1.$ Applying $\left(2.4 \right)$ along with triangle inequality for $Y_{2}$, we have

By setting $\left \vert \sigma _{1}\right \vert = \varkappa$ with $\varkappa \in \left(0, 1\right],$ we obtain

Clearly $N^{\prime }\left(\varkappa \right) \leq 0,$ $N\left(\varkappa \right)$ is a decreasing function of $\varkappa$, indicating that it achieves its maxima at $\varkappa = 0$, that is,

Therefore

and so the required proof is accomplished. □

Theorem 4.7. Let $f^{-1}$ be the inverse of $f\in \mathcal{S}_{3l, s}^{\ast }$ with series expansion $\left(1.2\right)$. Then

Proof. The determinant $\left \vert \hat{H}_{3, 1}\left(f^{-1}\right) \right \vert$ is described as follows:

From $\left(4.9\right)$–$\left(4.12\right)$, we easily write

The below inequality follows easily by using Lemma 2.1 with $\varrho = - \frac{4}{5}$ and $\varsigma = 0$

and also by virtue of Lemma 2.3, we have

Applying it and also using $\left \vert \sigma _{4}\right \vert \leq 1-\left \vert \sigma _{2}\right \vert ^{2}-\left \vert \sigma _{1}\right \vert ^{2},$ we achieve

where

But $E$ is an increasing function of the variable $\sigma$; consequently,

The function $E\left(0, t\right)$ reaches its maximum value in $\left[0, 1 \right]$ if $t = 1,$ so $E\left(0, t\right) \leq \frac{29}{10},$ which completes the proof. □

Conjecture 4.8. If the inverse of $f\in \mathcal{S}_{3l, s}^{\ast }$ is of the form $\left(1.2\right),$ then

This result is best possible.

5.   Initial coefficients for $\mathcal{SK}_{\exp }$

Next, we begin this section by determining the estimates of the first four initial coefficients for functions in the family $f\in \mathcal{SK}_{\exp }.$

Theorem 5.1. If the function $f\in \mathcal{SK}_{\exp }$ has the series form $\left(1.1\right)$, then

The equality is attained by the following extremal functions:

Proof. Let $f\in \mathcal{SK}_{\exp }.$ Then there exists a Schwarz function $w$ such that

Also, assuming that

Using $\left(1.1\right)$, we obtain

By some easy calculation and utilizing the series expansion of $\left(5.3\right)$, we achieve

Now by comparing $\left(5.4\right)$ and $\left(5.5\right),$ we have

Using $\left(2.2\right)$ in $\left(5.6\right)$, we obtain

Rearranging of $\left(5.7\right)$, we obtain

Applying $\left(2.3\right)$ in the above equation, we achieve

For $d_{4},$ we can write $\left(5.8\right)$, as

From Lemma 2.1, let

It is clear that $\frac{1}{2}\leq \left \vert \varrho \right \vert \leq 2$ with

Hence the conditions of Lemma 2.1 are satisfied. Therefore

From $\left(5.9\right)$, we deduce that

The initial segment is estimated by $\frac{1}{2}$ by utilizing $\left(2.7\right)$ with $\Lambda = 1$. Lemma 2.3 uses for the estimation of the second segment in the following:

By adding the bounds of the segments of $\left(5.10\right),$ we achieve

Thus, the proof is completed. □

6.   Inverse coefficients for $\mathcal{SK}_{\exp }$

Lastly, we will investigate the estimates of first three initial inverse coefficients for functions in the family $\mathcal{SK}_{\exp }.$

Theorem 6.1. If the inverse function of $f\in \mathcal{SK}_{\exp }$ is of the form $\left(1.2\right)$, then

Equalities hold in these bounds and will be confirmed by using $\left(1.3\right)$–$\left(1.5\right)$ and $\left(5.1 \right)$ with $m = 1, 2, 3.$

Proof. Applying $\left(5.6\right)$–$\left(5.9\right)$ in $\left(1.3\right)$–$\left(1.6\right)$, we achieve

Using $\left(2.2\right)$ in $\left(6.1\right)$, we obtain

For $B_{3},$ we can write $\left(6.2\right)$, as

Applying $\left(2.3\right)$ in the above equation, we achieve

For $B_{4},$ we consider

From Lemma 2.1, let

It is clear that $\frac{1}{2}\leq \left \vert \varrho \right \vert \leq 2$ with

This shows that all conditions of Lemma 2.1 are satisfied. Thus

Thus, the required proof is completed. □

Theorem 6.2. If $f\in \mathcal{SK}_{\exp }$ has inverse function $f^{-1}$ with a series form $\left(1.2\right),$ then

This inequality is sharp.

Proof. Employing $\left(6.1\right)$ and $\left(6.2\right)$, we have

Implementation of Lemma 2.2 along with triangle inequality leads us to

The functional bound is sharp and will be obtained from $\left(1.3 \right)$, $\left(1.4\right)$, and $\left(5.1\right)$ with $m = 2.$ □

By replacing $\tau = 1$ in Theorem 6.2, we arrive at the below result.

Corollary 6.3. If $f\in \mathcal{SK}_{\exp }$ has the inverse function with a series form $\left(1.2\right)$, then

The functional bound is sharp. Equality will be achieved by utilizing $\left(1.3\right)$, $\left(1.4\right)$, and $\left(5.1\right)$ with $m = 2.$

Theorem 6.4. If the inverse of the function $f\in \mathcal{SK}_{\exp }$ is expressed in $\left(1.2\right)$, then

This outcome is sharp, and it will be confirmed easily by using $\left(1.3\right)$–$\left(1.5\right)$, and $\left(5.1\right)$ with $m = 3.$

Proof. From $\left(6.1\right)$–$\left(6.3\right)$, we have

From Lemma 2.1, let

It is clear that $\frac{1}{2}\leq \left \vert \varrho \right \vert \leq 2$ with

Thus, all the conditions of Lemma 2.1 are satisfied. Hence

and hence the proof is completed. □

Theorem 6.5. If the inverse function of $f\in \mathcal{SK}_{\exp }$ is provided in $\left(1.2\right),$ then

Equality will be held by using $\left(1.3\right)$, $\left(1.5\right)$, and $\left(5.1\right)$ with $m = 3.$

Proof. Putting $\left(6.1\right)$ and $\left(6.3\right)$, we have

From Lemma 2.1, let

It is clear that $\frac{1}{2}\leq \left \vert \varrho \right \vert \leq 2$ with

Thus, all the conditions of Lemma 2.1 are satisfied. Hence

□

Theorem 6.6. Let $f^{-1}$ be the inverse of $f\in \mathcal{SK}_{\exp }$ as defined in $\left(1.2\right)$. Then

Equality will be achieved by using $\left(1.3\right)$–$\left(1.5\right)$ and $\left(5.1\right)$ with $m = 2.$

Proof. From $\left(6.1\right)$–$\left(6.3\right)$, we have

where

and

Utilizing Lemma 2.4, we obtain $R_{1}\leq 1.$ Also, by virtue of Lemma 2.3 for $R_{2},$ we achieve

Since $\left(-\frac{\left \vert \sigma _{1}\right \vert }{8\left(\left \vert \sigma _{1}\right \vert +1\right) }+1\right) > 0.$ Thus, we can substitute $\left(2.4\right)$ in $\left(6.5\right)$, and we easily obtain

The basic computation of maximum and minimum leads us to

Hence

The proof is thus accomplished. □

Theorem 6.7. Let $f^{-1}$ be the inverse function of $f\in \mathcal{SK} _{\exp }$ and is expressed in $\left(1.2\right)$. Then

Proof. The determinant $\left \vert \hat{H}_{3, 1}\left(f^{-1}\right) \right \vert$ can be expressed as follows:

From $\left(6.1\right)$–$\left(6.4\right)$, we easily write

At the beginning, it should be noted that

where we have used Lemma 2.1 with $\varrho = -\frac{7}{15}$ and $\varsigma = -\frac{1}{10}.$ Also, by using Lemma 2.3, we have

Applying it and also using $\left \vert \sigma _{4}\right \vert \leq 1-\left \vert \sigma _{2}\right \vert ^{2}-\left \vert \sigma _{1}\right \vert ^{2},$ we achieve

where

But $E$ is a decreasing function of the variable $\sigma;$ consequently,

The function $E\left(0, t\right)$ reaches its maximum value in $\left[0, 1 \right]$ if $t = -\frac{45}{512}+\frac{1}{512}\sqrt{100329},$ so $E\left(0, t\right) \leq 1.7079,$ which completes the proof. □

Conjecture 6.8. If the inverse function of $f\in \mathcal{SK}_{\exp }$ is of the form $\left(1.2\right)$, then the sharp bounds

Equality will be achieved by using $\left(1.3\right)$–$\left(1.5\right)$ and $\left(5.1\right)$ with $m = 3.$

7.   Conclusions

The study of Hankel determinant bounds is of great importance in the research community due to its vast applications in mathematical science. In the current article, we have considered the Hankel determinant involving the coefficients of inverse functions for various subclasses of analytic functions. This generalizes the classical definition of the Hankel determinant and could provide more knowledge of inverse functions. The main focus of this article that we have studied is the coefficient-related problems along with Hankel determinants for the inverse function of the functions that belong to the families of symmetric starlike and symmetric convex functions associated with three different image domains. In particular, these problems include the sharp estimates of some initial inverse coefficients, the Zalcman, Fekete–Szegö, and Krushkal inequalities, along with the sharp estimation of second and third Hankel determinants containing inverse coefficients for functions in the mentioned families by using the concept of a Schwarz function. Also, we have given some conjectures that strongly support our obtained results. Our research introduces a new framework for analyzing the Hankel determinant, emphasizing the importance of inverse coefficients in analytic functions, potentially promoting more attention to coefficient-related problems. This study may be applied to meromorphic analytic functions, and the same methodology can be used to examine higher-order Hankel determinants, as studied in articles ^[46,47,48].

Author contributions

Huo Tang: Funding acquisition, Methodology, Project administration; Muhammad Abbas: Investigation, Writing-original draft; Reem K. Alhefthi: Formal analysis, Supervision, Writing-review and editing; Muhammad Arif: Formal analysis, Supervision, Writing-review and editing. All authors read and approved the final manuscript.

Acknowledgments

The first author (Huo Tang) was partly supported by the Natural Science Foundation of China under Grant 11561001, the Program for Young Talents of Science and Technology in Universities of Inner Mongolia Autonomous Region under Grant NJYT18-A14, the Natural Science Foundation of Inner Mongolia of China under Grants 2022MS01004 and 2020MS01011, and the Higher School Foundation of Inner Mongolia of China under Grant NJZY20200, the Program for Key Laboratory Construction of Chifeng University (No. CFXYZD202004), the Research and Innovation Team of Complex Analysis and Nonlinear Dynamic Systems of Chifeng University (No. cfxykycxtd202005) and the Youth Science Foundation of Chifeng University (No. cfxyqn202133).

The third author (Reem K. Alhefthi) would like to extend their sincere-appreciation to the Researchers Supporting Project number (RSPD2024R802) King Saud University, Riyadh, Saudi Arabia.

Conflict of interest

The authors declare that they have no conflicts of interest.